Method for demodulating a signal integrating phase error effect and corresponding receiver

ABSTRACT

The invention concerns a method for demodulating a digital signal received via a transmission channel, comprising a step which consists in associating with each value received of said received signal a point of the corresponding modulation constellation, on the basis of the decision boundaries taking into account the potential effect of a phase shift on at least one of said points of the modulation constellation and of the potential effect of an Gaussian additive noise applied on said point, said Gaussian additive noise being represented by a generating surface associated with said point, and said phase shift by a rotation on an angular range based so that said swept surface belongs essentially to the region of decision associated with the corresponding point of the modulation constellation, plotted on the basis of at least one phase and/or amplitude characteristic of said modulation, so as to associate with each of said points of the constellation a portion of a reception space, called corresponding region of decision.

CROSS-REFERENCE TO RELATED APPLICATION

This Application is a Section 371 National Stage Application ofInternational Application No. PCT/FR02/01641, filed May 15, 2002 andpublished as WO 02/093862 on Nov. 21, 2002, not in English.

FIELD OF INVENTION

The domain of the invention is transmission of digital signals,particularly in the presence of phase noise. More precisely, theinvention relates to an improvement in demodulation of such signals, andparticularly optimisation of latching of the synchronisation system anda reduction in the probability of this synchronisation system becomingunlatched.

BACKGROUND OF THE INVENTION

The invention is used in applications in very many technical domains,for single-carrier and for multi-carrier signals, particularly foramplitude modulations in quadrature (MAQ) regardless of the number ofstates. It is particularly advantageous for transmission in burst mode.

Systems developed in telecommunications operate at increasingly highfrequencies, using modulations with a very large number of states. Thequality of the local oscillator that controls the frequencytransposition then becomes a technological lock. As the frequency ofthese systems increases, it becomes technologically more difficult todesign oscillators with good frequency stability and low phase noise.

Therefore an attempt is made to optimise performances of the phaselocking loop to overcome problems due to degradation to systemperformances induced by hyperfrequency oscillators of the type availableto the general public.

In general, demodulation consists of putting received values into aspace taking account of the modulation constellation used. This space isbroken down into decision regions, defined by decision-makingboundaries. Each region is assigned to one of the constellation stateswhich is considered to be the most probable, and that is selected as thedemodulation result when a received value is located in this region.

In the following, we will present examples of an MAQ modulation receivedin a single-sensor receiver. The reception space is then the Fresnelplane (I/Q plane). This two-dimensional example efficiently describesthe state-of-the-art and the characteristics of the invention. However,it is quite clear that the invention is equally applicable to othermodulation types, possibly using spaces with more than two dimensions.Similarly, the use of multi-sensor receivers can lead to the definitionof reception spaces with more than two dimensions.

Therefore, MAQ type digital modulation techniques are based on the useof a modulation constellation in single-sensor receivers, conventionallyrepresented in the I/Q plane in the form shown in FIG. 1 in the specialcase of an MAQ16 modulation (only the first quadrant is shown. The threeother quadrants are directly deduced by symmetry).

The modulation points 11 are uniformly distributed at equal distancesfrom each other. The modulation then consists of choosing one of thepoints 14 from among the 16 points available in the constellation. Thereceived value 12 after transmission through a transmission channelaffected by various disturbances is often significantly offset (13) fromthe ideal point 14.

Therefore, the demodulation operation consists of associating thereceived value 12 with the most probable emitted point 14. This is doneby defining demodulation boundaries 15 parallel to the I and Q axes,maximising the distances (the received value 12 is considered tocorrespond to the closest point 14). Therefore, these boundaries 15define zones 16, each associated with a point 14 in the modulationconstellation.

This technique provides a relatively efficient means of overcomingGaussian additive noise. On the other hand, errors can occur in thepresence of important phase errors, as is the case particularly in thesynchronisation system latching phase in the presence of a frequencyoffset, or in the presence of loud phase noise. For example, a phaseshift 17 will lead to a demodulation error, the received value 18 beingconsidered to correspond to point 19 and not to point 14.

In particular, one purpose of the invention is to overcome thisdisadvantage in the state of the art.

More precisely, one purpose of the invention is to provide ademodulation technique for reducing the effects of a frequency offsetmore efficiently than is possible with a conventional technique.

Consequently, one purpose of the invention is to provide such atechnique enabling faster latching of the synchronisation system,particularly in the presence of a frequency offset.

Obviously, another purpose is to provide such a technique for reducingthe probability of the synchronisation system becoming unlatched.

Another purpose of the invention is to provide such a technique that iseasy and inexpensive to implement, particularly in receivers used by thegeneral public without needing to make any modifications tohyper-frequency oscillators.

Another purpose of one particular aspect of the invention is to providesuch a technique that is adaptive, and that takes account of alldisturbances induced through the transmission channel (phase noise orGaussian additive noise).

These objectives, and others that will become clearer later, areachieved using a method for demodulation of a digital signal receivedthrough a transmission channel, comprising a step in which each receivedvalue of the said received signal is associated with a correspondingpoint in the modulation constellation, as a function of decision-makingboundaries, plotted as a function of at least one phase and/or amplitudecharacteristic of the said modulation, so as to associate acorresponding decision region with each of the said points in theconstellation.

SUMMARY OF THE INVENTION

According to the invention, the process comprises the following steps:

-   -   association of at least one generating zone enclosing the said        point with at least one of the said points in the said        modulation constellation, the zone representing the potential        effect of Gaussian additive noise;    -   application of a rotation to the said generating zone in the        said reception space, over an angular range that depends on        symmetries defined by the said modulation, so as to define a        surface scanned by the said generating zone, representing the        potential effect of a phase shift on the said point;    -   definition of at least one boundary, chosen such that the said        scanned surface belongs essentially to the decision region        associated with the corresponding point in the modulation        constellation.

Thus, the invention proposes to modify conventional modulationboundaries (usually minimising distances from points in the modulationconstellation), taking account firstly of the fact that under someconditions a phase error can significantly shift a received signal pointfrom the corresponding emitted point, and secondly the fact that thereceived signal may be disturbed by Gaussian additive noise (white noiseand/or coloured noise).

Therefore, it is proposed that this received point should notsystematically be assigned to the closest point in the constellation,but to the most probable point taking account of a potential phaseshift.

Note that according to this aspect, the invention does not require anyspecific processing at the emission (although one advantageousmodulation process will be proposed later on). Therefore, the samesignal may be processed firstly by conventional receivers, and secondlyand more efficiently in terms of the error rate and/or the latchingrate, by receivers using the demodulation process according to theinvention.

However, it will be noted that receivers implementing the invention takeaccount of aspects related to the emission (the structure of theconstellation used) and reception (Gaussian noise).

According to one preferred embodiment of the invention, the saiddecision-making boundaries are plotted in the I/Q plane so as toassociate a decision region corresponding to a portion of the said I/Qplane, with each of the said points in the modulation constellation.Obviously, the same approach can be adapted for other embodiments.

Preferably, in this case, the said boundaries are variable as a functionof variations in the said Gaussian additive noise. It is thus possibleto optimise demodulation as a function of reception conditions.

Advantageously, the said generating zone forms a disk, the radius ofwhich may for example be proportional to the standard deviation of thesaid Gaussian additive noise.

Preferably, at least one of the said disks is centered on thecorresponding point in the said modulation constellation.

Advantageously, at least two concentric generating zones are taken intoaccount, to trace at least one boundary for at least one of the saidpoints in the said modulation constellation.

According to one particular embodiment, at least one of the saidboundaries is a combination of at least one portion of a boundarycorresponding approximately to an edge of the said scanned surface andat least one linear portion corresponding to an axis of symmetry definedby the said modulation constellation.

According to one advantageous embodiment of the invention, at least oneof the said generating zones is not centered on the corresponding pointin the said modulation constellation, so as to simulate a modificationto the constellation at the emission.

The points associated with at least one boundary adapted as a functionof the potential effect of a phase shift preferably comprise at leastthe points in the constellation furthest from the centre of the said I/Qplane.

These are the points that are most sensitive to phase errors. Thereforein simplified embodiments, it can be assumed that they are the onlypoints concerned.

According to one preferred embodiment, the said modulation constellationcorresponds to an amplitude modulation in quadrature (MAQ).

In particular, boundaries like those shown in FIG. 5 or 11 or 13 areadvantageously used in the case of an MAQ modulation 16 (it isinconvenient and inefficient to describe these boundariesmathematically, while the figures give a direct understanding. This iswhy, exceptionally, reference is made to the figures in thecorresponding claim).

Depending on the specific embodiment, the said received signal may be amulti-carrier signal or a single-carrier signal. In particular, it maybe a signal transmitted in burst, in which case the invention isparticularly advantageous.

The demodulation process according to the invention is advantageouslyused during a latching phase in a phase locking loop.

It may also be used advantageously under continuous receptionconditions, after a phase locking loop has been latched, either at alltimes or at least in the presence of loud phase noise.

According to one preferred embodiment of the invention, it is plannedthat if the Gaussian additive noise is greater than a predeterminedthreshold, the said boundaries ignore the said potential effect of phasenoise. The result is a conventional constellation.

According to one particular embodiment, the process according to theinvention comprises the following steps:

-   -   compare the said received value with a first set of boundaries,        called conventional boundaries, formed so as to maximise        distances between the said points in the said constellation and        so as to make a first decision on the emitted point        corresponding to the said received value;    -   measure the amplitude of the received value with respect to the        centre of the said constellation;    -   measure the signal-to-noise ratio;    -   possibly modify the said first decision, as a function of the        said amplitude and the said signal-to-noise ratio, so as to        provide a second decision based on the said boundaries taking        account of the potential effect of a phase shift;    -   if applicable, lift the ambiguity between at least two points in        the said modulation constellation, as a function of a        measurement of the angular position of the said received value.

The invention also relates to a modulation process for a digital signalusing a modulation constellation, according to which the position of atleast one of the points in the said modulation constellation is chosentaking account of the potential effect of a phase rotation on thispoint, so as to increase the probability of the corresponding receivedvalue being correctly demodulated, after transmission through atransmission channel that could induce the said phase rotation.

Once again, the objective is to take account of the potential action ofa phase error. However, in this case this action is anticipated toobtain a better demodulation quality in reception.

It is possible, but not compulsory, to implement the modulation processand the demodulation process described above in the same system.

The invention also relates to receivers of a digital signal receivedthrough a transmission channel using the demodulation process describedabove. This type of receiver comprises demodulation means comprisingmeans of associating a corresponding point in the modulationconstellation with each received value of the said received signal, as afunction of decision-making boundaries plotted as a function of at leastone phase and/or amplitude characteristic of the said modulation, so asto associate each of the said points in the constellation with acorresponding decision region.

According to the invention, at least one of the said boundaries isadapted taking account firstly of the potential effect of a phase shifton at least one of the said points in the modulation constellation, andsecondly the potential effect of Gaussian additive noise applied to thesaid point, the said Gaussian additive noise being represented by agenerating surface associated with the said point, and the said phaseshift by a rotation on an angular range that depends on symmetriesdefined by the said modulation, so as to define a surface scanned by thesaid generating zone, the said boundary being chosen such that the saidscanned surface belongs approximately to the decision region associatedwith the corresponding point in the modulation constellation.

The invention also relates to a system for transmission of at least onedigital signal, from at least one emitter to at least one receiver,using means of modifying the modulation constellation on emission and oron reception, and/or means of modifying the correspondingdecision-making boundaries, taking account firstly of the potentialeffect of a phase shift on at least one of the said points in themodulation constellation, and secondly the potential effect of Gaussianadditive noise applied to the said point, the said Gaussian additivenoise being represented by a generating surface associated with the saidpoint, and the said phase shift by rotation on an angular range thatdepends on symmetries defined by the said modulation, so as to define asurface scanned by the said generating zone, the said boundary beingchosen such that the said scanned zone belongs mostly to the decisionregion associated with the corresponding modulation constellation point.

Finally, the invention also relates to a digital signal using amodulation constellation, in which the position of at least one of thepoints is chosen taking account of the potential effect of phaserotation on this point, so as to increase the probability of thecorresponding received value being correctly demodulated aftertransmission through a transmission channel that could induce the saidphase rotation.

BRIEF DESCRIPTION OF THE DRAWINGS

Other characteristics and advantages of the invention will become clearafter reading the following description of preferred embodiments of theinvention given as simple illustrative examples, and the attacheddrawings among which:

FIG. 1, already described in the preamble, illustrates a modulationconstellation MAQ16, and the principle of its demodulation according toprior art;

FIG. 2 shows a block diagram of a digital synchronisation system, knownin itself;

FIG. 3 illustrates the characteristic of the detector in FIG. 2 forE_(S)/N_(O)=19 dB, according to the technique used in prior art;

FIG. 4 is a general mimic diagram of an implementation of the invention;

FIGS. 5 a and 5 b show the first quadrant of a constellation MAQ16 usingmodified decision-making boundaries according to a first embodiment ofthe invention;

FIG. 6 illustrates an example use of demodulation using the boundariesin FIG. 5;

FIG. 7 illustrates a characteristic of a phase detector using thedecision-making boundaries in FIG. 5, for E_(S)/N_(O)=19 dB;

FIG. 8 shows a first quadrant of a constellation MAQ16 modified at theemission, according to the invention;

FIG. 9 compares tolerances to a phase error for the conventional MAQ16constellation and the constellation in FIG. 8;

FIG. 10 illustrates the characteristic of a phase detector forE_(S)/N_(O)=19 dB, when the constellation in FIG. 8 is used;

FIG. 11 presents the first reception quadrant of a constellation MAQ16modified as shown in FIG. 8 and with modified boundaries according tothe invention;

FIG. 12 illustrates the characteristics of a phase detector forE_(S)/N_(O)=19 dB, in the case of a decision based on FIG. 11;

FIG. 13 shows the first reception quadrant of a constellation MAQ16,using boundaries modified according to the invention and a simulation ofa modification in this constellation at the emission;

FIG. 14 illustrates the characteristic of a phase detector forE_(S)/N_(O)=19 dB in the case of a decision based on FIG. 13.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

1 —THE STRUCTURE OF THE SYNCHRONISATION SYSTEM

FIG. 2 shows an example embodiment consisting of a digital carriersynchronisation system of a receiver using a Directed Decision (DD)algorithm derived from application of a Maximum Likelihood (ML)criterion based on a feedback (FB) structure and prior retrieval of therate (T).

The structure of the system is based on the derivative of the Maximum APosteriori Likelihood criterion (1) with respect to the phase error.Thissystem is called DDMLFBT and is composed of three elements; a phasedetector 21, a feedback filter 22 and an integrator 23, as shown infigure 2.

Nevertheless, solutions according to the invention may be applicable inall digital carrier synchronisation systems based on a Directed Decisionalgorithm that uses a received symbols estimate.

We will not discuss details of other elements in this FIG. 2, which areknown in themselves. The emitted signal s(t) is received in the formr(t), after transmission through a transmission channel 24. Thisreceived signal is sampled (25) and then demodulated using a multiplier26 controlled by the integrator 23. The real part (27) and the imaginarypart (28) are separated from the demodulated signal w(k). They can beused to make a comparison with the original constellation (29, 210), andare input to the phase detector 21.

The role of the phase detector 21 in which we are particularlyinterested within the context of this invention, is to provideinformation representative of the phase error. This information is thenfiltered (22) and then integrated (23) in the loop in order to generatethe phase correction {circumflex over (θ)} to be made to the signal.

1.1 The Phase Detector.

The phase detector 21 is the keystone of the feedback structure and mustbe capable of evaluating the residual error between the samplew(k)=r(k)e^(−je) with phase correction and the estimated symbol{circumflex over (d)}(k) used as the phase reference. This estimatedsymbol is obtained by applying conventional decision-making boundariesF₀ of the constellation C₀ related to the modulation used, to the symbolw(k).

The phase detector 21 may be defined by its characteristic ε(φ) derivedfrom application of the Maximum Likelihood criterion and that forexample may be determined by the following expressions [2]:ε₁(k)=Im ^([{circumflex over (d)}*(k)w(k)])ε₂(k)=Im ^([csgn[w*(k)]w(k)])ε₃(k)=Im ^([w*(k)csgn[w(k)−{circumflex over (d)}(k)]])ε₄(k)=Im ^([csgn[w*(k)][w(k)−{circumflex over (d)}(k)]])ε₅(k)=Im^([{circumflex over (d)}*(k)csgn[w(k)−{circumflex over (d)}(k)]])ε₆(k)=Im ^([csgn[w*(k)]csgn[w(k)−{circumflex over (d)}(k)]])

where csgn(x)=sgn[Re[x]+jsgn[Im[x]]

A study of the characteristics of phase detectors made by D. Mottier [1]leads to selection of the detector ε₄(k) for its good properties in thecase of MAQ type modulations. Therefore, this detector is used as anexample in the following, associated with an MAQ16. However, the processdescribed below is equally applicable regardless of the type of detectorchosen and regardless of the order of the constellation MAQ.

The characteristic of the selected detector associated with an MAQ16 fora signal-to-noise ratio E_(S)/N_(O)=19 dB is shown in FIG. 3. Thedecision-making device used to generate the estimated symbols{circumflex over (d)}(k) uses conventional decision-making boundaries F₀of the constellation C₀ relative to the MAQ16. This characteristicreveals the following intrinsic properties of the phase detector:

-   -   its period:

${{ɛ(\phi)} = {ɛ\left( {\phi + {k\frac{\pi}{2}}} \right)}},$kεZ. This property is the result of invariance of the MAQ constellationat a phase rotation of

$k{\frac{\pi}{2}.}$Among other things, this makes it possible to study only a singlequadrant of the modulation used;

-   -   its false latch points: none. There is a false latch point when        the output from the detector cancels out and the sign of its        slope is the same as the sign of the slope at the origin, while        the phase error is not zero;    -   its linear range 31: 0.2 radians (11.5 degrees). Within the        linear range at the origin of the characteristic, the detector        outputs information ε(φ) representative of the phase error.        Thus, as the length of the linear phase increases, the detector        becomes increasingly capable of detecting a large phase error.        Therefore, this provides a means of reducing the probability of        the synchronisation system becoming unlatched in the presence of        phase noise. Furthermore, the size of the linear range        determines the feedback latching capacity in the presence of a        frequency offset;    -   its gain K_(d)=1.2. The gain of the detector is defined as being        the slope of the linear range at the origin. As the value of        K_(d) increases, the value of ε(φ) increasingly represents        univocal information representative of the phase error.

The phase detector is sensitive to the noise level of the input signal.When the noise increases, its linearity range and its gain decrease. Onthe other hand, in some cases noise minimises the probability of falselatching points.

Loop Characterisation

Assuming that the gain K_(d) of the detector and the gain K₀ of theintegrator are normalised, the estimated phase update relation iswritten as follows:

${\hat{\theta}\left( {k + 1} \right)} = {{\hat{\theta}(k)} + {\alpha\;{ɛ(k)}} + {\beta{\sum\limits_{jsk}{ɛ(j)}}}}$

where α and β are the positive coefficients of the feedback filter.

In general, carrier recuperation systems use a second order feedbackstructure [3]. This is why this structure is used in examples describedbelow, although once again this use is not restrictive.

In this case, the closed loop transfer function can be expressed in thefollowing form:

${Q(z)} = \frac{z^{- 1}\left( {{\alpha\left( {1 - z^{- 1}} \right)} + \beta} \right.}{\left( {1 - z^{- 1}} \right)^{2} + {z^{- 1}\left( {{\alpha\left( {1 - z^{- 1}} \right)} + \beta} \right.}}$

The structure of the second order feedback loop can be defined by twoparameters more significant than α and β. The damping factor ξ is astability parameter determining oscillations of the estimated phasecurve {circumflex over (θ)}(k). It is usually assumed that ξ=0.707[4],to guarantee that the loop is stable. Furthermore, the parameter used isthe equivalent monolateral noise band of the loop B_(j) that isnormalised with respect to the duration of the symbols T_(S). Thelatching velocity increases as the value of B_(j)T_(S) increases, butthe loop also generates a noisier estimate {circumflex over (θ)}(k).

The expression for B₁T_(S) is defined as follows:

${B_{1}T_{s}} = {\frac{1}{2}{\int_{- \infty}^{+ \infty}{{{Q(f)}}^{2}\ {\mathbb{d}f}}}}$

The coefficients α and β are deduced from loop parameters as follows:

$\alpha = {\frac{2B_{j}T_{s}}{\xi + \frac{1}{4\xi}}\left( {{2\;\xi} - \frac{B_{j}T_{s}}{\xi + \frac{1}{4\xi}}} \right)}$$\beta = \left( \frac{2B_{j}T_{s}}{\xi + \frac{1}{4\xi}} \right)^{2}$

Performances of the conventional solution using a decision-making devicebased on constellation C₀ and the decision-making boundaries F₀ inlatching mode are given in table 1 for E_(S)/N_(O)=19 dB. Latching timeswere measured in the case of a frequency offset Δƒ₀=134 kHz and fordifferent values of the equivalent normalised noise band B₁T_(S). As wehave already mentioned, the latching time reduces as B₁T_(S) increases.

TABLE 1 Performances of the conventional synchronisation system in thepresence of a frequency offset Δƒ₀ = 134 kHz and for E_(s)/N_(o) = 19dB. Decision-making device B_(l)T_(s) = 5 × 10⁻³ B_(l)T_(s) = 1 × 10⁻²B_(l)T_(s) = 5 × 10⁻² C₀ and F₀ 745000 T_(s) 53000 T_(s) 360 T_(s)Presentation of the Invention

Examples about modifications use the parameters in table 2, so as topresent homogeneous numeric results. Obviously, this is a non-limitativeexample.

TABLE 2 parameters used in application examples. Parameter ValueDetector ε₄ (k) Modulation MAQ16 Symbol throughput l/T_(s) = 6.8 MS/sSignal-to-noise ratio E_(s)/N₀ = 19 dB2.1 First Embodiment: Modification of Decision-Making Boundaries

Tolerance to a phase error can be improved by modifying decision-makingboundaries, at least for some symbols in the constellation C₀.Advantageously, any modification to decision-making boundaries is theresult of a compromise between tolerance to Gaussian noise and to aphase error.

2.1.1 Principle

2.1.1.1 General Principle (FIG. 4)

FIG. 4 shows a simplified block diagram illustrating the generalprinciple of an embodiment of the invention.

At least one generating zone (55) representative of the potential effectof Gaussian additive noise on the point considered (see FIG. 5) isassociated (41) with each point in the constellation (or at least withsome points, and in this case preferably at least the external points ofthe constellation).

This generating zone 55 could be a circle, but other shapes could alsobe considered. In this case, the radius of the circle is advantageouslya function of the standard deviation σ of the Gaussian additive noise42. In other words, the system is adaptive as a function of the Gaussiannoise level 42 (obviously, in a simplified version the boundaries couldbe fixed to correspond to an average situation).

Information on additive noise can be obtained by various known methods,and for example by analysis of the signal received during a periodduring which no signal is emitted or during which a reference signal(known to the receiver) is transmitted.

Several generating zones 56, 57 (FIG. 5) (for example two, correspondingto circles with radii σ and 2σ) are advantageously taken into account,at least for some of the points, to optimise the boundaries.

They may or may not be concentric.

The generating zones may be concentrated on the point in theconstellation or they may be offset from it (third embodiment).

Once these generating zones have been obtained, a rotation 58 is applied(43) to them, so as to define a scanned surface 59 representative of thepotential effect of a phase rotation. Since this rotation is applied tothe generating zone, the scanned surface is representative firstly ofthe effect of Gaussian additive noise and secondly the effect of a phaserotation.

The rotation range applied to each of the generating zones depends onsymmetries induced by the constellation. Thus, referring to the examplein FIG. 5, points 51 and 52 are affected by a rotation of π/2. On theother hand, points 53 and 54, which are both on the same radius, arerotated by π/4.

The result is thus a series of plots of portions of circles 5101, 5102,5103 corresponding to the edges of scanned surfaces.

Starting from these elements, adaptive boundaries 5101, 5102, 5103 aredefined (44) that enable more efficient demodulation in the presence ofphase noise and therefore particularly better latching of thesynchronisation system. Thus, for example, the received value 512 willbe coffectly associated with point 52, while according to theconventional technique it would be incoffectly associated with point 53.

The boundaries are formed starting from arcs of circles 5101, 5102,5103, portions of straight lines 5131, 5132 corresponding to mediatingplanes between points, or symbols in the constellation.

Obviously, these boundaries may be slightly modified. For example, itcould be decided to linearise all or some of the arcs of circles, ifthis simplifies the implementation.

Detailed Example (FIG. 5)

FIG. 5 illustrates the embodiment of this type of compromise in the caseof a signal-to-noise ratio E_(S)/N_(O)=19 dB. The symbols of the firstquadrant of the constellation C₀ are represented by points (+a, +a),(+3a, +a), (+a, +3a) and (+3a, +3a) where a=1/√{square root over (10)}is the energy normalisation factor.

In order to take account of the phase error in the presence of Gaussianadditive white noise (BBAG) to estimate received symbols, we definedecision regions delimited by arcs of circles and mediating planesbetween symbols located at the same distance from the centre of theconstellation. These new zones are the result of the displacement ofsymbols around a circle in the presence of a phase error.

For example, the radius of circles centered on constellation symbols arer=σ and r=2σ, where σ is the standard deviation of Gaussian additivenoise (other values of the type α.σ could be used) . The probabilitythat a symbol affected by Gaussian noise is within the circle withradius σ is of the order of 90%. Thus, decision-making boundaries areadapted such that the tolerance to a phase error is maximum for allnoisy symbols contained within the circle with radius a or within thecircle with radius 2σ, if this is possible.

It is found that the modified boundaries particularly affect decisionsmade about symbols external to the constellation that are the mostsensitive to phase errors.

However, note that there is a limit to the application of thisprinciple: the maximum value of the standard deviation of Gaussian noisemust be less than a/2 (where 2a is the minimum distance betweensymbols). This application limit is represented by a minimumsignal-to-noise ratio of 16 dB in the case of an MAQ16.

2.1.3 Example Embodiment

The implementation of a demodulation based on this principle can bebroken down into two distinct parts, as shown in FIG. 6.

The first step consists of a conventional demodulation 61 (according toFIG. 1) which associates the symbol {circumflex over (d)}(k) of theclosest constellation C₀ with a received symbol w(k); this is equivalentto making a decision with respect to conventional boundaries F₀.

The second step consists of applying an algorithm 62 that will bedenoted M_(A), making a second decision starting from the result of theconventional demodulation {circumflex over (d)}(k) and the receivedsymbol w(k). This algorithm uses mapping 63 of the constellation and thesignal-to-noise ratio 64 as parameters. With these two parameters, asecond decision can then be made about the received symbol w(k) by usingthe modified decision-making boundaries relative to the first estimatedsymbol {circumflex over (d)}(k) denoted F₀M_(A) and shown as acontinuous line in FIG. 5 (5103, 5132, 5101, 5131, 5102).

In practice, it is more judicious to perform this procedure in two stepssince in the second step, it is necessary to consider boundariesmodified according to algorithm M_(A) and only related to the symbol{circumflex over (d)}(k) estimated during the first step.

All that has to be taken into account is the amplitude of the receivedvalue, and if necessary the phase shift of this value (if there is anyambiguity between two possible symbols with the same amplitude).

The result of this operation outputs a final estimated symbol{circumflex over (d)}_(M)(k). If the received symbol (wk) belongs to themodified decision region of the first estimated symbol {circumflex over(d)}(k) then {circumflex over (d)}_(M)(k)={circumflex over (d)}(k) else{circumflex over (d)}_(M)(k)≢{circumflex over (d)}(k).

2.1.4 Detector Characteristic

The characteristic of the detector that uses the estimated symbols{circumflex over (d)} (k) output from the modified decision-makingdevice (C₀ F₀ M_(A)) is shown in FIG. 7 for E_(S)/N₀=19 dB.

It can be seen that the. linear range for the proposed solution (3radians, or 17.2 degrees) is slightly greater than the linear range fora conventional solution (2 radians, or 11.5 degrees).

2.1.5 Performances

Table 3 shows latching mode performances of the synchronization systembased on the modified decision-making device that uses the constellationC₀ associated with decision-making boundaries F₀M_(A). Theseperformances were obtained for a frequency offset Δƒ₀=134 kHz, a signalto noise ratio E_(S)/N₀=19 dB and for different values of B₁T_(S).

It is found that the modification to boundaries used by thedecision-making device reduces the latching time by a factor equal to2.5 for B₁T_(S)=5×10⁻² and 4.5 for B₁T_(S)=5×10⁻³.

TABLE 3 Performances of the modified synchronization system (C₀ andF₀M_(A)) in the presence of a frequency offset Δƒ₀ = 134 kHz and forE_(s)/N₀ = 19 dB. Decision-making device B_(l)T_(s) = 5 × 10⁻³B_(l)T_(s) = 1 × 10⁻² B_(l)T_(s) = 5 × 10⁻² C₀ and F₀ 745000 T_(s) 53000T_(s) 360 T_(s) C₀ and F₀M_(A) 162000 T_(s) 11300 T_(s) 136 T_(s)

2.2 Second Embodiment: Modification of the Constellation on Emission

2.2.1 Principle

The inventors noticed that if the external symbol is translated fromposition (+3a, +3a) to position (+(3+x)a, +(3+x)a), the tolerance to aphase error associated with this symbol can be increased. Similarly, bytranslating the cross symbols from positions (+3a, +a) and (+a, +3a) tothe corresponding positions (+(3−y)a, +a) and (+a, +(3−y)a), thetolerance to a phase error associated with these symbols is improved.The inventors confirmed that the values of x and y must satisfy thefollowing condition for it to be possible to work at a constantnormalization factor a=A/√{square root over (10)}:6x+x ²=6y−y ²

The Appendix contains the corresponding demonstration.

For small values of x and y, this relation may be approximated by x≈y.In any case, we will choose small values so as not to excessivelydegrade performances in the presence of Gaussian additive noise. Forreadability reasons, we will identify the conventional constellationwith the label C₀ and the modified constellation displayed in FIG. 8with the label C₁. The constellation C₁ was determined by using x=y=0.1.Therefore, it is defined by symbols 81 to 84 in its first quadrant (+a,+a), (+2.9a, +a), (+a, +2.9a) and (+3.1a, +3.1a). It will be seen thatthe new positions of the symbols lead to a slight modification of thedecision-making boundaries 85 that will be denoted F₁, in opposition tothe conventional boundaries F₀ of a constellation C₀.

FIG. 9 represents the tolerances to phase errors of the differentsymbols in a conventional constellation C₀ and the modifiedconstellation C₁. It shows that tolerances are better in the case of theconstellation C₁.

2.2.2 Characteristic of the Phase Detector

The characteristic of the detector that uses estimated symbols{circumflex over (d)}(k) output from the modified decision-making device(C₁, F₁) is shown in FIG. 10 for E_(S)/N₀=19 dB.

The proposed solution has a larger linear range (2.39 radians or 13.7degrees) than a conventional solution (2 radians, or 11.5 degrees).

2.2.3 Performances

Table 4 shows the performances in latching mode of the synchronizationsystem based on the modified decision-making device that uses themodified constellation C₁ and its relative decision-making boundariesF₁. These performances were obtained by a frequency offset Δƒ₀=134 kHz,a signal-to-noise ratio E_(S)/N₀=19 dB and for different values ofB₁T_(S).

TABLE 4 Performances of the notified synchronization system (C₁ and F₁)in the presence of the frequency offset Δƒ₀ = 134 kHz and for E_(s)/N₀ =19 dB Decision-making device B_(l)T_(s) = 5 × 10⁻³ B_(l)T_(s) = 1 × 10⁻²B_(l)T_(s) = 5 × 10⁻² C₀ and F₀ 745000 T_(s) 53000 T_(s) 360 T_(s) C₁and F₁ 405000 T_(s) 42000 T_(s) 300 T_(s)

It is found that the modification to the constellation used by thedecision-making device provides a means of reducing latching times by afactor of between 1.2 for B₁T_(S)=5×10⁻² and 1.8 for B₁T_(S)=5×10⁻³.

2.3 Third Embodiment: Combinations of Previous Solutions (Modificationof the Constellation and Decision-Making Boundaries)

Performances can be improved by combining the previous two optimisationsdescribed above: modification of decision-making boundaries andmodification of the constellation.

2.3.1 First Variant

2.3.1.1 Principle

A first possible variant of the modified demodulation is a combinationof a modified constellation C₁ of decision-making boundaries F₁ and amodified boundaries algorithm M_(A). The first quadrant of such aconstellation is shown in FIG. 11 in the case of an E_(S)/N₀ ratio equalto 19 dB. The. resulting decision-making boundaries 111 will be denotedF₁M_(A).

2.3.1.2 Detector Characteristic

FIG. 12 shows the characteristic of the detector that uses estimatedsymbols d(k) output from the modified decision-making device (C₁,F₁M_(A)) for E_(S)/N₀=19 dB.

It can be seen that the linear range for the proposed solution (2.89radians or 16.6 degrees) is more than the linear range for aconventional solution (2 radians, or 11.5 degrees).

2.3.1.3 Performances

Table 5 shows performances in latching mode of the synchronizationsystem based on the modified decision-making device that uses themodified constellation C₁ and the modified decision-making boundariesF₁M_(A). These performances were obtained for a frequency offset Δƒ₀=134kHz, a signal-to-noise ratio E_(S)/N₀=19 dB, and for different values ofB₁T_(S).

It can be seen that a modification of the constellation used by thedecision-making device provides a means of reducing latching times by afactor of between 3 for B₁T_(S)=5×10⁻³ and 3.5 for B₁T_(S)=5×10⁻².

TABLE 5 Performances of the modified synchronization system (C₁ andF₁M_(A)) in the presence of a frequency offset Δƒ₀ = 134 kHz and forE_(s)/N₀ = 19 dB Decision-making device B_(l)T_(s) = 5 × 10⁻³ B_(l)T_(s)= 1 × 10⁻² B_(l)T_(s) = 5 × 10⁻² C₀ and F₀ 745000 T_(s) 53000 T_(s) 360T_(s) C₁ and F₁M_(A) 241000 T_(s) 24500 T_(s)  98 T_(s)

2.3.2 Second Variant

2.3.2.1 Principle

The second variant uses a constellation C₁ combined with an algorithmthat we will denote M_(B). This algorithm is different from thealgorithm M_(A) in that it uses a virtual constellation and not theconstellation used, as a parameter. The effect of this virtualconstellation is to centre the circles with the radii σ and 2σ onvirtual symbols, which induces a modification to the decision-makingboundaries obtained when the algorithm M_(A) is used. The virtualconstellation provided as a parameter is composed of the followingsymbols (+a, +a), (+2.8a, +a), (+a, +2.8a) and (+3.2a, +3.2a). Thedecision-making boundaries 131 used are shown in FIG. 13.

2.3.2.2—Detector Characteristic

The characteristic of the detector that uses the estimated symbols^(·)(k) output from the modified decision-making device (C₁, F₁M_(B)) isshown in FIG. 14 for E_(S)/N₀19 dB.

It is observed that the linear range for the proposed solution (2.89radians or 16.6 degrees) is more than the linear range for aconventional solution (2 radians, or 11.5 degrees).

2.3.2.3—Performances

Table 6 shows the performances in latching mode of the synchronizationsystem based on the modified decision-making device that uses themodified constellation C₁ and the modified decision-making boundariesF₁MB. These performances were obtained for a frequency offset Δƒ₀=134kHz, a signal-to-noise ratio E_(S)/N₀=19 dB and for different values ofB₁T_(S).

TABLE 6 Performances of the modified synchronization system (C₁ andF₁M_(B)) in the presence of a frequency offset Δƒ₀ = 134 kHz and forE_(S)/N₀ = 19 dB Decision-making device B_(l)T_(s) = 5 × 10⁻³ B_(l)T_(s)= 1 × 10⁻² B_(l)T_(s) = 5 × 10⁻² C₀ and F₀ 745000 T_(s) 53000 T_(s) 360T_(s) C₁ and F₁M_(B) 249000 T_(s) 17900 T_(s)  98 T_(s)

It is found that the modification to the constellation used by thedecision-making device provides a means of reducing latching times by afactor of between 3 for B₁T_(S)=5×10⁻³ and 5 for B₁T_(S)=5×10⁻².

2.4—Summary of the Modifications Made

2.4.1—Detector Characteristics

The dimensions of the linear ranges of the phase detector related to theassociated decision-making devices are given in Table 7.

TABLE 7 Size of linear ranges of the phase detector Decision-makingdevice Size of linear range C₀ and F₀ 11.5 degrees C₁ and F₁ 13.7degrees C₀ and F₀M_(A) 17.2 degrees C₁ and F₁M_(A) 16.6 degrees C₁ andF₁M_(B) 16.5 degrees

2.4.2—Performances

Table 8 shows PLL performances in acquisition mode for the differentdecision-making devices studied, in the case of a frequency offsetΔƒ₀=134 kHz as a function of the equivalent PLL noise band B₁ normalizedas a function of the symbol throughput 1/T_(S)=6.8MS/s.

TABLE 8 Performances in acquisition mode for E_(S)/N₀ = 19 dB fordifferent demodulation types used by the DDMLFBT and for differentvalues of B_(l)T_(S.) Modulation type B_(l)T_(S) = 5 × 10⁻³ B_(l)T_(S) =1 × 10⁻² B_(l)T_(S) = 5 × 10⁻² C₀ and F₀ 745000 T_(s) 53000 T_(s) 360T_(s) C₁ and F₁ 405000 T_(s) 42000 T_(s) 300 T_(s) C₀ and F₀M_(A) 162000T_(s) 11300 T_(s) 136 T_(s) C₁ and F₁M_(A) 241000 T_(s) 24500 T_(s)  98T_(s) C₁ and F₁M_(B) 249000 T_(s) 17900 T_(s)  98 T_(s)

The simulation results show a significant reduction in the latching timein the case in which modified decision-making devices are used,regardless of the equivalent noise band used.

As long as B₁T_(S) remains less than 10⁻², the (C₀, F₀M_(A)) solutionappears to be the most attractive. On the other hand, the (C₁, F₁M_(A))and (C₁, F₁M_(B)) solutions can give better latching times for highervalues of B₁T_(S).

Moreover, a study was carried out on performances of the differentconfigurations in tracking mode. It was observed that performances wereidentical when the (C₀, F₀), (C₁, F₁), (C₁, F₁M_(A)) and (C₁ F₁M_(B))decision-making devices were used. On the other hand, the performancesof the synchronization system associated with the (C₀, F₀M_(A))decision-making device are slightly less optimised than the previousfour solutions in tracking mode.

3. Optimisation of Demodulation Functions

The decision-making devices described above were also used in thedemodulation system. In this part, we will present the performances onthe Gaussian channel of the demodulator MAQ16 associated with differentdecision-making devices, and if a local oscillator affected by a phasenoise is used. The noisy signal input to this demodulator after thecarrier has been retrieved is affected by a residual phase error with acentered Gaussian probability density and variance σ_(ε) ². Table 9presents the performances obtained in terms of bit error rates forE_(S)/N₀=19 dB and for different values of the variance of the phaseerror that existed before demodulation.

TABLE 9 Performances for E_(s)/N₀ = 19 dB in the presence of a residualGaussian phase error with variance σ_(ε) ² σ₂ ^(ε) C₀ and F₀ C₁ and F₁C₀ and F₀M_(A) C₁ and F₁M_(B) 4 × 10⁻¹ 2.78 × 10⁻¹ 2.75 × 10⁻¹ 2.40 ×10⁻¹* 2.50 × 10⁻¹  1 × 10⁻¹ 1.31 × 10⁻¹ 1.24 × 10⁻¹ 1.04 × 10⁻¹* 1.06 ×10⁻¹  4 × 10⁻² 5.12 × 10⁻² 4.55 × 10⁻² 3.78 × 10⁻²* 3.82 × 10⁻²  1 ×10⁻²  4.6 × 10⁻³  3.5 × 10⁻³ 3.11 × 10⁻³* 3.12 × 10⁻³  8 × 10⁻³  2.8 ×10⁻³  2.1 × 10⁻³ 1.9 × 10⁻³  1.8 × 10⁻³* 5 × 10⁻³  1.0 × 10⁻³  7.6 ×10⁻⁴ 7.8 × 10⁻⁴  7.3 × 10⁻⁴* 1 × 10⁻³  1.0 × 10⁻⁴  8.9 × 10⁻⁵* 1.7 ×10⁻⁴  1.0 × 10⁻⁴  5 × 10⁻⁴  6.5 × 10⁻⁵*  6.6 × 10⁻⁵ 1.5 × 10⁻⁴  8.3 ×10⁻⁵   1 × 10⁻¹²  4.2 × 10⁻⁵*  4.8 × 10⁻⁵ 1.2 × 10⁻⁴  6.9 × 10⁻⁵ 

On each line of this table, the * symbol adjacent to a value of thevariance of the phase error indicates the decision-making device thathas the best performances.

This results table shows three possible configurations:

-   -   for large variances, the modified decision-making devices have        the best performances;    -   for moderate to weak variances, use of the constellation C₁ is a        good compromise;    -   as would be expected, the lowest BER for very low variance        values is obtained by the conventional decision-making system.

Demodulation performances were also studied in the case of a largesignal-to-noise ratio E_(S)/N₀=30 dB. These results presented in Table10 demonstrate that the improvement in performances made by the use ofmodified decision-making devices is particularly significant when thesignal-to-noise ratio is high.

TABLE 10 Performances for E_(s)/N₀ = 30 dB in the presence of a residualGaussian phase error with variance σ_(ε) ² σ₂ ^(ε) C₀ and F₀ C₁ and F₁C₀ and F₀M_(A) C₁ and F₁M_(B) 4 × 10⁻¹ 2.75 × 10⁻¹ 2.72 × 10⁻¹ 2.37 ×10⁻¹* 2.51 × 10⁻¹ 1 × 10⁻¹ 1.22 × 10⁻¹ 1.15 × 10⁻¹ 9.60 × 10⁻²* 1.01 ×10⁻¹ 4 × 10⁻² 4.12 × 10⁻² 3.49 × 10⁻² 2.82 × 10⁻²* 3.17 × 10⁻² 1 × 10⁻² 6.8 × 10⁻⁴ 4.02 × 10⁻⁴ 3.10 × 10⁻⁴* 3.70 × 10⁻⁴ 8 × 10⁻³ 2.06 × 10⁻⁴1.03 × 10⁻⁴  7.8 × 10⁻⁵*  9.6 × 10⁻⁵ 5 × 10⁻³  7.6 × 10⁻⁶  2.9 × 10⁻⁶ 2.1 × 10⁻⁶*  2.6 × 10⁻⁶

4—Summary

The principles used for optimisation of the carrier recuperation anddemodulation system have been presented for the case of an MAQ16 and aDDMLFB-T system.

However, these principles can be applied to any amplitude modulation inquadrature with an order of more than four, and to any Directed Decisioncarrier recuperation system.

Note also that in the case of systems affected by strong Gaussian noise,it is always possible to modify decision-making boundaries related toexternal symbols of the constellation. These symbols are more sensitiveto phase errors, consequently this simple change to the boundariesprovides a means of significantly improving demodulation andsynchronization functions of the system in the presence of phase errors.

APPENDIX Normalisation of Energy for an MAQ16

Conventional MAQ16

Symbols in the first quadrant are (+a,+a), (+3a, +a), (+a, +3a). Tonormalize energy of symbols in constellation at 1, the value of a thatsolves the following equation has to be determined:(a ² +a ²)=((3a)²+(3a)²)+2((3a)² +a ²)=4hence2a ²+18a ²+20a ²=4and finally

$a = \frac{1}{\sqrt{10}}$

Modified MAQ16

Consider the case of a modified MAQ16 such that the symbols in the firstquadrant are (a, a), (a, a(3−y)), (a(3−y), a) and (a(3+x), a(3+x)). Wewill determine the required value of y when x is known, such that thevalue of a is identical to the value used in the case of a conventionalMAQ16. We then need to solve the following equation:2a ²+18a ²+20a ²+2a ²[6x+x ²−6y+y ²]=4

To keep the value a for a conventional MAQ16, we need to choose x and ysuch that the term between square brackets is zero. This means finding asolution to the following equation:6x+x ²=6y−y ²

Example: the value of y in the case of an external symbol fixed at(+3.1a, +3.1a), in other words for x=0.1, is y=0.103.

1. Method for demodulation of a digital signal received through atransmission channel, the method comprising: associating each receivedvalue of the received signal with a corresponding point in themodulation constellation, as a function of decision-making boundaries,plotted as a function of at least one phase and / or amplitudecharacteristic of the modulation, so as to associate a correspondingdecision region of a reception space with each of the points in theconstellation, said constellation being represented in an I/Q planehaving I and Q axes, wherein each of four quadrants of said I/Q planecomprises a set of points, comprising at least one external point, whichis the furthest from the center of said I/Q plane, and for each of saidexternal points, constructing demodulation boundaries defining adecision region associated with said external point, comprising thefollowing steps: association with said external point of at least onegenerating zone enclosing said external point with at least one of thepoints in the modulation constellation, the zone representing thepotential effect of Gaussian additive noise; application of a rotationof π/2, limited by the I and Q axes, to the generating zone in thereception space, so as to define a surface scanned by the generatingzone, representing the potential effect of a phase shift on saidexternal point; definition of at least one boundary including a firstarc of a circle from said I axis and a second arc of a circle from saidQ axis, chosen such that said first and second arcs of circle correspondto said scanned surface.
 2. The method of claim 1, wherein the decisionmaking boundaries are variable as a function of variations in theGaussian additive noise.
 3. The method of to claim 1, wherein the diskis centered on the corresponding external point generating zone forms adisk.
 4. The method of claim 3, characterized in that the radius of thedisk is proportional to the standard deviation of the Gaussian additivenoise.
 5. The method of claim 3, wherein the disk is centered on thecorresponding external point in the modulation constellation.
 6. Themethod of claim 1, wherein at least one of the decision makingboundaries is a combination of at least one portion of a boundarycorresponding approximately to an edge of the scanned surface and atleast one linear portion corresponding to an axis of symmetry defined bythe modulation constellation.
 7. The method of claim 1, wherein at leastone of the generating zones is not centered on the correspondingexternal point in the modulation constellation, so as to simulate amodification to the constellation at the emission.
 8. The method ofclaim 1, wherein the points associated with at least one decision makingboundary adapted as a function of the potential effect of a phase shiftpreferably comprise at least the points in the constellation furthestfrom the center of the reception space.
 9. The method of claim 1,wherein the modulation constellation corresponds to an amplitudemodulation in quadrature.
 10. The method of claim 9, wherein thereceiver is a single-sensor receiver, and the reception space is theFresnel plane.
 11. The method of claim 1, wherein the received signal isa multi-carrier signal.
 12. The method of claim 1, wherein the receivedsignal is a single-carrier signal.
 13. The method of claim 1, whereinthe received signal is transmitted in burst.
 14. The method of claim 1,wherein it is used during a latching phase in a phase locking loop. 15.The method of claim 1, wherein it is used under continuous receptionconditions, after a phase locking loop has been latched, at least in thepresence of loud phase noise.
 16. The method of claim 1 ,wherein in thepresence of a Gaussian additive noise greater than a predeterminedthreshold, the decision making boundaries ignore the potential effect ofphase noise.
 17. The method of claim 1, wherein it comprises thefollowing steps: compare the received value with a first set ofboundaries, called conventional boundaries, formed so as to maximizedistances between the points in the constellation and so as to make afirst decision on the transmitted point corresponding to the receivedvalue; measure the amplitude of the received value with respect to thecenter of the constellation; and measure the signal-to-noise ratio. 18.Receiver of a digital signal received through a transmission channel,comprising demodulation means comprising means of associating acorresponding point in the modulation constellation with each receivedvalue of the received signal, as a function of decision-makingboundaries plotted as a function of at least one phase and / oramplitude characteristic of the modulation, so as to associate each ofthe points in the constellation with a corresponding decision region ofa reception space, said constellation being represented in an I/Q planecomprising I and Q axes, each of four quadrants of said I/Q planecomprising a set of points, comprising at least one external point,which is the furthest from the center of said I/Q plane, characterizedin that the receiver comprises means for constructing, for each of saidexternal points, demodulation boundaries defining a decision regionassociated with said external point comprising means for modifying atleast one of the boundaries, taking account firstly of the potentialeffect of a phase shift on at least one of the external points in themodulation constellation, and secondly the potential effect of Gaussianadditive noise applied to the external point, the Gaussian additivenoise being represented by a generating surface associated with saidexternal point, and the phase shift by a rotation of π/2, limited by theI and Q axes, so as to define a surface scanned by the generating zone,the boundary being chosen such that the scanned surface belongsapproximately to the decision region associated with the correspondingpoint in the modulation constellation, and including a first arc of acircle from said I axis and a second arc of a circle from said Q axis,chosen such that said first and second arcs of circle correspond to saidscanned surface.
 19. System for transmission of at least one digitalsignal, from at least one emitter to at least one receiver, wherein ituses means for modifying the modulation constellation on emission, saidconstellation being represented in an I/Q plane comprising I and Q axes,each of four quadrants of said I/Q plane comprising a set of points,comprising at least one external point, which is the furthest from thecenter of said I/Q plane, and means for modifying the correspondingdecision-making boundaries on reception, taking account firstly of thepotential effect of a phase shift on at least one of the external pointsin the modulation constellation, and secondly the potential effect ofGaussian additive noise applied to said external point, the Gaussianadditive noise being represented by a generating surface associated withsaid external the point, and the phase shift by rotation of π/2, limitedby the I and Q axes, so as to define a surface scanned by the generatingzone, the boundary being chosen such that the scanned surface belongsmostly to the decision region associated the corresponding externalpoint in the modulation constellation, and including a first arc of acircle from said I axis and a second arc of a circle from said Q axis,chosen such that said first and second arcs of circle correspond to saidscanned surface.
 20. The method of claim 17, further including:modifying the first decision, as a function of the amplitude and thesignal-to-noise ratio, so as to provide a second decision based on thedecision making boundaries taking account of the potential effect of aphase shift; and overcoming any ambiguity between at least two externalpoints in the modulation constellation, as a function of a measurementof the angular position of the received value.